The Ring Chime device and the Ring Chime Pro device currently make the same sounds for Ring (doorbell sound) and Motion-detection, when you have multiple Ring devices connected to the same/single Chime device. So, if you have two doorbells (or some combinations of a doorbell and other Ring devices) connected to a single Chime, they all will produce the same sound from the Chime, and you cannot differentiate (from the sound alone) whether it is the front or rear door.
Others have been requesting the feature that you are looking for. If you like the idea for Chime/Chime-Pro to emit separate sounds from several Ring devices connected to a single Chime device, you should make your vote be heard by clicking on the all the links below, and then click on the “Thumbs Up” icon on the beginning thread (upper-left side) of the originator’s ‘Feature-Request-Board’. More “Thumbs Up” Kudo vote will get the attention of the Ring Teams. I’ve probably missed a few links in the “Feature-Request-Board” but you can see you aren’t alone!